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Properties

Label 2-3872-8.5-c1-0-32
Degree 22
Conductor 38723872
Sign −0.460−0.887i-0.460 - 0.887i
Analytic cond. 30.918030.9180
Root an. cond. 5.560405.56040
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.44i·3-s + 0.200i·5-s − 2.34·7-s − 2.99·9-s − 1.63i·13-s − 0.490·15-s + 3.33·17-s + 4.92i·19-s − 5.74i·21-s + 9.25·23-s + 4.95·25-s + 0.0187i·27-s − 6.19i·29-s + 8.25·31-s − 0.470i·35-s + ⋯
L(s)  = 1  + 1.41i·3-s + 0.0896i·5-s − 0.886·7-s − 0.997·9-s − 0.452i·13-s − 0.126·15-s + 0.809·17-s + 1.12i·19-s − 1.25i·21-s + 1.92·23-s + 0.991·25-s + 0.00360i·27-s − 1.14i·29-s + 1.48·31-s − 0.0795i·35-s + ⋯

Functional equation

Λ(s)=(3872s/2 ΓC(s) L(s)=((−0.460−0.887i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3872s/2 ΓC(s+1/2) L(s)=((−0.460−0.887i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38723872    =    25⋅1122^{5} \cdot 11^{2}
Sign: −0.460−0.887i-0.460 - 0.887i
Analytic conductor: 30.918030.9180
Root analytic conductor: 5.560405.56040
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3872(1937,⋅)\chi_{3872} (1937, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3872, ( :1/2), −0.460−0.887i)(2,\ 3872,\ (\ :1/2),\ -0.460 - 0.887i)

Particular Values

L(1)L(1) ≈\approx 1.7352733931.735273393
L(12)L(\frac12) ≈\approx 1.7352733931.735273393
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
11 1 1
good3 1−2.44iT−3T2 1 - 2.44iT - 3T^{2}
5 1−0.200iT−5T2 1 - 0.200iT - 5T^{2}
7 1+2.34T+7T2 1 + 2.34T + 7T^{2}
13 1+1.63iT−13T2 1 + 1.63iT - 13T^{2}
17 1−3.33T+17T2 1 - 3.33T + 17T^{2}
19 1−4.92iT−19T2 1 - 4.92iT - 19T^{2}
23 1−9.25T+23T2 1 - 9.25T + 23T^{2}
29 1+6.19iT−29T2 1 + 6.19iT - 29T^{2}
31 1−8.25T+31T2 1 - 8.25T + 31T^{2}
37 1+1.28iT−37T2 1 + 1.28iT - 37T^{2}
41 1−6.03T+41T2 1 - 6.03T + 41T^{2}
43 1−2.47iT−43T2 1 - 2.47iT - 43T^{2}
47 1+4.75T+47T2 1 + 4.75T + 47T^{2}
53 1−8.76iT−53T2 1 - 8.76iT - 53T^{2}
59 1−1.90iT−59T2 1 - 1.90iT - 59T^{2}
61 1+6.39iT−61T2 1 + 6.39iT - 61T^{2}
67 1−8.27iT−67T2 1 - 8.27iT - 67T^{2}
71 1−1.24T+71T2 1 - 1.24T + 71T^{2}
73 1+6.89T+73T2 1 + 6.89T + 73T^{2}
79 1+14.6T+79T2 1 + 14.6T + 79T^{2}
83 1+9.37iT−83T2 1 + 9.37iT - 83T^{2}
89 1−5.11T+89T2 1 - 5.11T + 89T^{2}
97 1−0.299T+97T2 1 - 0.299T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.927927460168701049133121448405, −8.121605156716974550802924286271, −7.27397437679017290417707601487, −6.32731908730038543500743171228, −5.67213612949106150729717921875, −4.84657244703713285074725600707, −4.16326052534573386235827978771, −3.21287246999243114949563357912, −2.86398880180400874648921069394, −1.01792149932073417670137532645, 0.64651758553240247448092062488, 1.43592350844067539980809300216, 2.74650799214331671168817715862, 3.15365183197612233377950750041, 4.56648536146156241677488966266, 5.34393032233311443490062718877, 6.36226811192891505175994701007, 6.88150340957450235821382834622, 7.20072061904479536738483201083, 8.199992100355681390658070289998

Graph of the ZZ-function along the critical line